The rain wets you in two different ways: by falling on your head and shoulders and by colliding with your front as you walk forward. The amount of rain that lands on your head and shoulders is directly proportional to how long you are in the rain, but the amount of rain that hits your front depends only on the density of the raindrops. Your speed has no effect on the amount of rain that hits your front, but it definitely affects the amount of rain landing on you from above. To minimize the rain dropping onto you, you should walk as fast as you can.

The microprocessor does everything in the navigation system except for activities that require specialized, dedicated electronic circuitry. For a unit based on the global positioning system (GPS), there are specialized circuits to detect and time microwave signals emitted by an array of earth-orbiting satellites and specialized circuits to operate a display and a sound system, but that’s about it. Everything else is done by one or more microprocessors.

Microprocessors now play a dominant role in most electronic devices because microprocessors allow their sophisticated circuits to be reused endlessly for many different purposes. Prior to the microprocessor revolution, electronic devices had to implement separate circuitry for each and every task or process. Nothing could be reused or reassigned, even if it wasn’t doing anything most of the time. With microprocessors, reuse and reassignment is easy and leads to enormous improvements in circuit efficiency. Devices that were once huge, expensive, and power-hungry can now be reduced to a small printed circuit board featuring a tiny microprocessor and a handful of dedicated circuits that are responsible for conveying information to and from that microprocessor.

A navigation system is a fine example of this shift to microprocessor-centric electronics. Earlier navigation systems, such as the instrument navigation systems (VOR) used by aircraft, involved a lot of specialized electronics and still only provided the pilot with information about the plane’s angle relative to a ground-based transmitter. The pilot needed at least two of these systems and a set of good maps to figure out where the plane was.

A modern, GPS-based navigation system simply hands the pilot or driver complete information about where the plane or car is located. It does this using specialized input circuitry and specialized but very ordinary computer output circuitry (a display and sound system). Everything else is done by the microprocessor.

The input circuitry in a GPS receiver is interesting. The global positioning system consists of an array of earth-orbiting satellites that use microwaves to announce their locations and emit timing pulses. The navigation system’s circuitry detects those microwaves, determines the satellites locations, and uses the timing pulses to measure the distance to each satellite it can detect. Since microwaves travel at just under the speed of light through earth’s atmosphere, knowing how long it takes for a pulse to travel from a satellite to the navigation system is equivalent to measuring the distance separating those two objects.

Once the navigation system’s specialized electronics has received information about the satellites’ locations and measured the travel times of their pulses, that information is passed along to the microprocessor. The microprocessor does everything else. It converts pulse travel times into distances, uses those distances to triangulate the navigation system’s location relative to the satellites, looks up information about that location in its stored maps, and displays relevant information to the pilot or driver. Easy peasy.

The specialized circuitry necessary to receive location and timing information from the GPS satellites was once complicated and expensive. But the desire to put such circuitry in every cellphone propelled amazing miniaturization. The GPS receivers now built into cellphones are tiny and inexpensive, and they include their own microprocessors. Part of the microprocessor revolution has been that many devices contain entire networks of microprocessors rather than just one centralized microprocessor. Having many microprocessors that communicate via a network can save a lot of specialized wiring and it turns out that wires can be more expensive than microprocessors. Cars and planes now have communications networks that allow their many microprocessors to exchange information and control components with only a handful of wires.

A floating object is displacing fluids that would otherwise fill the space it occupies. For example, a ball floating motionless on water is displacing the water and air that would normally be where the ball is. If we remove the ball, water and air will fill its space and soon everything will be motionless again.

Just because that ball-shaped portion of water and air is motionless doesn’t mean that it’s weightless. It does have a weight! But its weight is supported by the water and air that surround it. Because of the earth’s gravity, the pressure of stationary water or air decreases steadily with altitude, so pressure exerted on the bottom of this ball-shaped portion is greater than the pressure exerted on its top. This unbalanced pressure produce a net upward force on the ball-shaped portion of water and air.

That upward force is known as the buoyant force and it’s evidently just strong enough to support the weight of the ball-shaped portion of water and air. If it weren’t the ball-shaped portion would accelerate up or down.

When we put the real ball back where it was and let it again float motionless on the water, the surrounding water and air continue to exert the same buoyant force on the real ball that they exerted on the ball-shaped portion of water and air. So the ball experiences an upward buoyant force that’s equal in amount to the weight of the water and air it displaces. That observation is known as Archimedes’ principle.

Which brings me to your question. Here are two identical balls floating motionless on fresh water (left) and on salt water (right). In each case, the ball is experiencing a buoyant force that exactly cancels its weight. To obtain that exact buoyant force, the ball must displace a portion of water and air that weighs exactly as much as the ball weighs.

Salt water is denser than fresh water, meaning that salt water has more mass per volume (more kilograms per liter) than fresh water. A liter of salt water consequently weighs more than a liter of fresh water. Displacing a liter of salt water therefore produces a stronger upward buoyant force than displacing a liter of fresh water. That’s why the ball is floating higher on the container of salt water than it does on the container of fresh water.

The ball doesn’t need to displace as much salt water to obtain a buoyant force that supports its weight, so it rises higher on the salt water than it does on the fresh water. In each case, the ball finds just the right mix of water and air so that it displaces exactly its own weight in those two fluids.

The earth receives heat from the sun at an incredible rate and, if it didn’t get rid of that heat, it would get hotter and hotter. To maintain a steady average temperature, the earth must radiate away heat just as fast as it receives that heat from the sun. In other words, the thermal radiation that the sun emits into empty space must be equal to the thermal radiation the earth receives from the sun.

Though they’re equal in amount, these two thermal radiations have quite different spectrums. Because the sun is extremely hot, its thermal radiation spectrum is largely visible light. The earth, on the other hand, is relatively cool and its thermal radiation spectrum is almost entirely invisible infrared light. While the sun’s thermal radiation is brilliantly visible, we can’t see the earth’s thermal glow or where it’s coming from, which is important because some of it comes from our atmosphere.

If the earth had no atmosphere, its average surface temperature would be approximately -18 °C. At that temperature, the earth’s surface would emit just enough thermal radiation to balance the thermal radiation it receives from the sun. If the earth were hotter, it would radiate away more heat than it receives and cool down. If the earth were cooler, it would radiate away less heat than it receives and warm up.

But the earth does have an atmosphere and that atmosphere contributes to these exchanges of thermal radiation. Although it’s relatively transparent to visible light, the atmosphere is able to emit and absorb significant portions of the infrared spectrum. As a result, a substantial fraction of the earth’s thermal radiation originates in its atmosphere.

Because of the atmosphere’s contribution to the earth’s thermal radiation, the average altitude at which the earth’s thermal radiation originates is not ground level. Instead, it’s 5 kilometers above sea level, the altitude at which the air temperature is about -18 °C. So the earth’s atmosphere shifts the -18 °C average radiating surface from sea level to an altitude of 5 kilometers above sea level.

If you’ve ever traveled up into the mountains and felt the air during your trip, you’ve probably noticed that air’s temperature decreases with altitude. The earth’s atmosphere has a natural temperature gradient of about -6.6 °C per kilometer upward. A simple explanation for that temperature gradient is that at higher altitudes, air must commit more of its overall energy to gravitational potential energy (energy stored in the force of gravity), leaving it with less for thermal energy (energy associated with temperature).

Since the air temperature increases by 6.6 °C each time you descend 1 kilometer, the air temperature at an altitude of 4 kilometers is -11.4 °C (-18 °C + 6.6 °C), at an altitude of 3 kilometers is -4.8 °C (-11.4 + 6.6 °C), …, and at sea level is 15 °C. Sure enough, the average historical air temperature at sea level is about 15 °C.

Which brings us to the question itself: “Why do greenhouse gases warm the earth?”

The 5 kilometer average altitude of origin for the earth’s thermal radiation actually depends on the atmosphere’s chemical composition. Some molecules in the air, notably nitrogen and oxygen, are remarkably transparent in the infrared and barely contribute to the earth’s thermal radiation. Other molecules, notably water, carbon dioxide, methane, and nitrogen oxides, interact strongly with infrared light and contribute significantly to the earth’s thermal radiation. Those thermally radiating gases are collectively known as “greenhouse gases.” The higher the concentration of those greenhouse gases in the atmosphere, the more the atmosphere contributes to the earth’s thermal radiation and the higher the average altitude of origin for the earth’s thermal radiation.

As human-produced greenhouse gases accumulate in the earth’s atmosphere, the average altitude of origin for the earth’s thermal radiation increases. If that average altitude were to rise from 5 kilometers to 6 kilometers, the average temperature at sea level would increase by another 6.6 °C to 21.6 °C. A rise of that magnitude would be catastrophic or even apocalyptic.

Alas, the average altitude of origin for the earth’s thermal radiation has already risen significantly since the industrial revolution and with it, a rise in the average temperature at sea level. The rate of temperature rise is alarming and the task of halting it or at least slowing it substantially cannot be put off for another generation. Even with serious international effort, it’s likely that many areas of the world will become uninhabitable by the next century, either because they are too hot for human survival or because they are under water as the result of rising sea levels.

Why is it easier for a bicycle to stay upright when it is moving faster? And hard to stand up straight when it is moving slowly? What forces are acting on it that depend on its speed?

A bicycle is an example of an object that is unstable at rest but stable in motion. That it is unstable at rest means that it has an unstable equilibrium: when it is exactly upright, it is at equilibrium (zero net force and therefore inertial). As soon as it tips even a tiny bit, however, forces and torques arise that push it away from that equilibrium. Since it spontaneously accelerates away from equilibrium, given a chance, that equilibrium is termed “unstable.” In contrast, a tricycle has a stable equilibrium — when tipped, forces and torques arise that push it back toward equilibrium and it spontaneously accelerates toward equilibrium.

Even though an equilibrium is unstable, it is possible to keep the object at equilibrium is some other mechanism acts to return it to that equilibrium. In the case of a broom balanced on your hand, the mechanism is you — you can keep the broom in its vertical unstable equilibrium if you move your hand around properly. The bicycle does that return-to-equilibrium trick for you automatically whenever the bicycle is moving forward . The bicycle steers itself automatically back to the the unstable equilibiurm. Taster it is moving forward, the more effective its automatic steering mechanism becomes.

How can a spinning object keep constant velocity with the direction of its parts changing at every instant?

When you consider an object as rotating, you normally stop thinking of its parts as moving in their own independent ways and treat the whole assembly as a single object. While it’s true that the various parts of that object are accelerating in response to internal forces those parts exert on another, the object as a whole is doing a simpler motion: it’s rotating about some axis. This ability to focus on a simple motion in the midst of countless complicated motions is an example of the beautiful simplifications that physics allows in some cases.

When you push on a rotating object, when are you doing work?

That’s an interesting question and requires two answers. First, if you push on a part of the rotating object and that part moves a distance in the direction of the force you exert, then you do work on it. In principle, it is possible to identify all the work that you do on the rotating object via this approach.

However, it is also possible to determine the work you do entirely in terms of physical quantities of rotation. If you exert a torque on the rotating object and it rotates the an angle in the direction of your torque, you again do work on the object. That’s the rotational version of the work formula: whereas force time distance is the translational work formula, torque times angle is the rotational work formula.

An important complication arises, however, in that you must measure the angle in the appropriate units: radians. The radian is the natural unit of angle and is effectively dimensionless (no units after it). When you multiple the torque times the angle in radians, the resulting units are those of work and energy. If you use a non-natural unit of angle, such as the degree, then you’ll have to deal with presence of the angle unit in your result.

When an object is rotating, both the “up” and “down” directions point along the vertical axis. Do they correspond to clockwise and counterclockwise?

Yes. Distinguishing between the two opposite directions of rotation using words alone requires that everyone agree on what to call those two directions. It also requires that everyone have an artifact that they can use to identify which direction is which. When something is spinning about a vertical axis, a carousel or merry-go-round, for example, then physicists name the two possible directions of rotation “up” and “down” and use a right-hand rule to identify which is which. Since most people have a right hand and know which hand it is, the necessary artifact is built-in.

In more common language, the two directions might be called “clockwise as viewed from above” and counter-clockwise as viewed from above”. In this case, the artifact is an old-fashioned analog clock and is probably more of a remembered artifact than one that is in the room with you. Nonetheless, that common naming convention is fine; it’s just wordier than the physicist’s version.

If you push someone on a swing with 50 N of force and they push back with 50 N of force, then why does the person still move? Shouldn’t they stay motionless if all the forces are cancelled out?

You’re struggling with the most common misconception about Newton’s third law of motion — the law stating that for every force object A exerts on object B, there is an equal but oppositely directed force exerted by object B on object A. The pair of forces described in Newton’s third law always act on different objects and therefore never cancel one another. Object A’s force on object B acts only on object B and can cause object B to accelerate. Object B’s force on object A acts only on object A and can cause object A to accelerate.

When you push someone on the swing, your force on the person affects that person and will affect their swinging motion. The person does indeed push back equally hard on you, but that force affects you! If you are wearing roller skates, you will accelerate backward and drift away from the swing.

Is deceleration something new or just acceleration in the opposite direction?

Deceleration is simply a special case of acceleration. An object decelerates by accelerating in the direction opposite its current velocity. For example, if your car is heading toward Washington at 60 mph (100 km/h) and you push on the brake pedal, your car will begin to accelerate in the direction pointing away from Washington and your forward velocity (toward Washington) will decrease with time. Since an object that accelerates in the direction opposite its velocity always slows down, it has become conventional to say that it is decelerating.